Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453948 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,19,15)(16,20,27,23)(17,22,18,25)(21,28,24,26);(1,2,4,3)(6,9)(10,11)(13,15,19,14)(16,24,27,21)(17,22,18,25)(20,26,23,28);(2,3)(5,7)(6,10)(8,12)(9,11)(13,16)(14,20)(15,23)(17,28)(18,26)(19,27)(21,25)(22,24);(1,3,4,2)(7,12)(10,11)(13,16)(14,21)(15,24)(17,26)(18,28)(19,27)(20,25)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=efi,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366492);